Drivers Club of Arizona sponsors educational events that provide high$-$performance driving instruction on actual race tracks. Because safety is a primary consideration at such events, many drivers choose to install roll bars in their cars. Bigelow Industries manufactures two types of roll bars for race cars:

$$ Model X$1$ is bolted to the car using existing holes in the car's frame. Each roll bar requires $20$ pounds of a special high alloy steel, $70$ manufacturing time, and $60$ minutes of assembly time.

$$ Model X$2$ is a heavier roll bar that must be welded to the car's frame. It requires $25$ pounds of the special high alloy steel, $150$ minutes of manufacturing time, and $40$ minutes of assembly time.

Bigelow$$s steel supplier indicated that at most $20,000$ pounds of the high$-$alloy steel will be available next quarter. In addition, they estimate that $1,750$ hours of manufacturing time and $800$ hours of assembly time will be available next quarter. The profit per unit amounts are $$200$ for model X$1$ and $$400$ per unit for model X$2.$

$2$a$)$ Formulate this problem as a linear programming model, starting with the definition of decision variables, and followed by the objective function and the constraints.

$2$b$)$ Find the optimal solution using the graphical method $($on a graph paper$).2$c$)$ Identify all constraints as binding or non$-$binding.

$2$d$)$ Interpret the optimal solution you identified.